Question: 23. In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency \omega(t) and average amplitude A(t) of the system change with time t. Which one of the following options schematically depicts these changes correctly?
(1) [diagram]
(2) [diagram]
(3) [diagram]
(4) [diagram]
Answer: Option (2)
Explanation:
For a spring–mass system, the angular frequency is
\omega = \sqrt{\frac{k}{m}}As sand leaks out, the total mass m decreases.
Since \omega \propto \frac{1}{\sqrt{m}},
a decrease in mass causes the frequency to increase with time.
Thus the correct graph for \omega(t)
must be an increasing curve.
Next, the amplitude decreases with time because the system loses energy due to
continuous leakage of sand (a non-conservative process).
With less mass and energy loss, the oscillation becomes progressively weaker, so
A(t) \ \text{decreases with time}Therefore, the correct option must show:
• \omega(t) increasing
• A(t) decreasing
Only Option (2) matches this behavior.